This paper presents a new regularization scheme for Digital Image Correlation (DIC) and Digital Volume Correlation (DVC) techniques based on the equilibrium gap method with reference to a linear elastic behavior. This scheme constitutes a unique framework for performing the so-called mechanical regu-larization for any problem dimension. "Complete regularization" refers to the fact that a specific treatment of boundaries (surfaces) is introduced here on the same footing as the bulk, independently of the complexity of their shape. The proposed treatment distinguishes the roles that different boundaries (Neumann or Dirichlet) play in mechanical tests. Numerical cases on synthetic data and a real experimental test validate the robustness and accuracy of the method. The analyzed experiment shows that only the use of (complete) regularization ensures convergence. Even in the cases where such regularization is not employed but convergence is achieved, it is at much higher cost. These results reveal the benefit of regularization on the convergence rate of DVC.